Finite Group Databasebeta

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#72259926

order5184
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

FittingC22×C33 ⋊ GL(2,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC2C2×C3⋊AGL(2,𝔽3)
commutator, cosocleC2×C3×PSU(3,𝔽2)⋊C3C22
FittingC22×C33GL(2,𝔽3)
socleC2×C33C2×GL(2,𝔽3)
hypercenterC22C3⋊AGL(2,𝔽3)
hypocenterC3×PSU(3,𝔽2)⋊C3D4
LCSG ▷ C2×C3×PSU(3,𝔽2)⋊C3 ▷ C3×PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2 ◁ C22

799 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 16 points:

(2,3)(5,7)(8,11,16,12,9,15,14,10);(4,5)(6,7)(9,10,14)(11,13,15);(1,2,3)

Polycyclic presentation:

2^3,3^2,2^3,3^2|f^2=h,g^2=h,b^a=bc,d^b=d^2,e^b=e^2,f^b=gh,f^e=fg,g^b=fh,g^e=f,g^f=gh,i^b=j,i^e=j,i^f=ij,i^g=ij^2,i^h=i^2,j^b=i,j^e=i^2j^2,j^f=ij^2,j^g=i^2j^2,j^h=j^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259971
maximal quotient of#72259971

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 8, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 5, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 6, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 7, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 8, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);